Bound and ground states for a class of Schrödinger–Poisson systems

AbstractWe are concerned with the following Schrödinger–Poisson system: 
 {−Δu+u+K(x)ϕu=a(x)u3,x∈R3,−Δϕ=K(x)u2,x∈R3.$$ \textstyle\begin{cases} -\Delta u+u+K(x)\phi u=a(x)u^{3},& x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, & x\in \mathbb{R}^{3}. \end{cases} $$ Assuming that K(x)$K(x)$ and a(x)$a(x)$ are nonnegative functions satisfying 
 lim|x|→∞a(x)=a∞>0,lim|x|→∞K(x)=0,$$ \lim_{|x|\rightarrow \infty }a(x)=a_{\infty }>0, \qquad \lim _{|x|\rightarrow \infty }K(x)=0, $$ and other suitable conditions, we show the existence of bound and ground states via a global compactness lemma and the Nehari manifold. Our result extends the existence result of positive solutions for Schrödinger–Poisson system with more than three times growth by Cerami and Vaira (J. Differ. Equ. 248:521–543, 2010) to the system with three times growth.


Introduction and main result
In this paper we are devoted to the following Schrödinger-Poisson system where the potentials K and a satisfy: (a 1 ) a ∈ L ∞ (R 3 ), lim |x|→∞ a(x) = a ∞ > 0.

System (SP) is a special form of the Schrödinger-Poisson system
(1. 1) which has been first introduced in [5] as a physical model describing a charged wave, interacting with its own electrostatic field in quantum mechanics. The unknowns u and φ represent the wave functions associated to the particle and electric potential, the functions V and K are external potentials, and the nonlinearity f (x, u) simulates the interaction between many particles or external nonlinear perturbations. For more information on the physical aspects, we refer the reader to [5].
There have been many works dealing with (1.1) and the existence of bound and ground states, radial and nonradial solutions, sign-changing solutions, and semiclassical states.
Here we mainly are interested in the existence results of bound and ground states. It is well known that the main difficulty of problem (1.1) is the lack of compactness for Sobolev's embedding theorem on the whole space R 3 . To recover the compactness, many studies were focused on the autonomous case (see [4, 9-11, 14, 17], for example) or on the radially symmetric function space which possesses compact embedding (see [3,15,17], for instance). When the potentials are neither constants nor radially symmetric, system (1.1) has also been increasingly receiving interest in recent years, for example, see [1][2][3]13]. Azzollini and Pomponio [4] considered the existence of ground states of (1.1) when V is nonconstant and possibly unbounded below, K = 1, f (x, u) = |u| p-1 u and f (x, u) = u 5 + |u| p-1 u with 3 < p < 5. In [23], Zhao and Zhao established the existence of ground states of (1.1) with pure power nonlinearity f (u) and V being weakly differentiable, K = 1. Later, Cerami and Vaira [8] considered the system where 3 < p < 5. Assume that the potential K(x) satisfies condition (K) and a(x) satisfies (a 1 ) lim |x|→∞ a(x) = a ∞ > 0, α(x) := a(x)a ∞ ∈ L 6 5-p (R 3 ); they discussed the existence of ground and bound states. More precisely, when (a 1 ), (a 2 ), (K) and other suitable restrictions on K(x) and a(x) are satisfied, they showed that problem (1.2) has a ground state by the standard method of Nehari manifold and compactness lemma. Moreover, when (a 1 ), (a 3 ), (K), and certain restrictions on K(x) and a(x) are satisfied, they proved that system (1.2) possesses a bound state via the linking theorem. After that some researchers focused on problem (1.2). Sun et al. in [19] generalized the existence results of ground states to the asymptotically linear case. In [13], the authors dealt with the equation system (1.4) When V ∈ L 3 2 (R 3 ), K ∈ L 2 (R 3 ) are nonnegative functions, and |V | 3 2 + |K| 2 is suitably small, they showed that (1.4) has a bound state via a linking theorem.
It seems that the existence of bound and ground states for problem (1.2) with p = 3, namely system (SP), has not been studied. So in this paper we will fill this gap. Compared with [8], the main difficulty is the lack of the higher-order term and the competing effect of the nonlocal term with three times growth term. On the one hand, due to the lack of the higher-order term, the standard method of Nehari manifold is invalid. Inspired by [12], by restricting the functional in a set, this functional has a unique maximum point along the nontrivial direction u in H 1 (R 3 ). Then we use the one-to-one correspondence of the functionals on the manifold and an open set of the unit sphere to establish the new method of Nehari manifold. On the other hand, the competing effect of the nonlocal term K(x)φu and three times growth term a(x)u 3 makes some estimations and verifications more complex.
Since we have no symmetry assumptions, similar to [8] we shall recover the compactness of PS sequences by the problem at infinity: In order to state the main results, we give some notations as follows. Denote by w the unique radial solution of (NSE) ∞ , and set m ∞ = 1 4 w 2 , where · is the standard norm of H 1 (R 3 ).
In addition, when (a 2 ) holds, the problemu + u = a(x)u 3 possesses a ground state w a , whose energy is m a = 1 4 w a 2 < m ∞ . We denote by S andS the best constants for the embeddings of H 1 (R 3 ) and D 1,2 (R 3 ) in L 6 (R 3 ), respectively. Our results are as follows: , (a 2 ) and (K) hold. Furthermore, assume either then system (SP) has a ground state.
Theorem 1.2 Let (a 1 ), (a 3 ) and (K) hold. In addition, assume then the system (SP) has a bound state.
The paper is organized as follows. In Sect. 2, we give some preliminaries. In Sect. 3, we introduce the variational setting. In Sects. 4 and 5, we prove Theorems 1.1 and 1.2, respectively.

Notations and preliminaries
In this paper we use the following notations: is the Sobolev space endowed with the scalar product and norm is denoted by | · | r ; S andS are the best Sobolev constants for the embeddings of H 1 (R 3 ) and D 1,2 (R 3 ) in L 6 (R 3 ), respectively, namely Without loss of generality, in the sequel, we assume that a ∞ = 1. System (SP) can be easily transformed into a Schrödinger equation with a nonlocal term. Actually, for all u ∈ H 1 (R 3 ), consider the linear functional L u defined in D 1,2 (R 3 ) by By Hölder and Sobolev inequalities, we have Hence, the Lax-Milgram theorem implies that there exists a unique φ u ∈ D 1,2 (R 3 ) such that Substituting φ u into the first equation of (SP), we get By (2.1) and (2.2), we obtain In addition, one easily has that the functional is of class C 1 and its critical points are solutions of (SP) . Define the operator Φ : Below we summarize some properties of Φ, which were proved in [6,8].

Variational setting
In this section we describe the variational framework for our problem. Firstly we give the Nehari manifold M corresponding to I: Hence I| M is coercive.
Next we introduce a set to construct the new method of Nehari manifold as in [21]. Define We claim that We shall argue as in [21] to show (3.2), but we need some modifications since K ∈ L 2 (R 3 ) in this paper, which is different than L ∞ (R 3 ) considered in [21]. Firstly, we need the following inequality: Proposition 3.1 (Hardy-Littlewood-Sobolev inequality; see [16]) Let s, r > 1, 0 < μ < 3 with 1 s + μ 3 + 1 r = 2, f ∈ L s (R 3 ), and h ∈ L r (R 3 ). There exists a sharp constant C(s, μ, r), independent of f and h, such that Hence we can choose r be so large that u 0 ∈ Θ. Then (3.2) follows. Set (iii) For each compact subset W of Θ ∩ S 1 , there exists C W > 0 such that t w ≤ C W for all w ∈ W . (iv) There exists ρ > 0 such that inf S ρ I > 0 and then Proof The proof is similar to that given in [21], and we state it for the reader's convenience.
(i) For each u ∈ Θ, one easily has that h(t) > 0 when t is sufficiently small, and h(t) < 0 when t is large enough. Then h has a positive maximum point in (0, ∞). Moreover, the maximum point t satisfies Then the maximum point is unique, and denoted by t u . Therefore, conclusion (i) follows.
(iii) Suppose that there exist a compact subset W ⊂ Θ ∩ S 1 and a sequence {w n } ⊂ W such that t w n → ∞. Assume w ∈ W is such that w n → w in H 1 (R 3 ). Then one easily has that However, by (3.1), we know that I(t w n w n ) ≥ 0. This is a contradiction.
(iv) By (2.3), one easily has that there exists ρ > 0 such that inf S ρ I > 0. For any u ∈ M, there is t > 0 such that tu ∈ S ρ . Note that I(u) ≥ I(tu), then inf S ρ I ≤ inf M I = c. Hence c > 0.
In addition, (v) easily follows from (3.1). This ends the proof. In order to restore the compactness for the PS sequence of (SP) , we introduce two equations. If K(x) = 0, then (SP) turns out to be and the Nehari manifold is When a(x) = a ∞ = 1, (NSE) becomes The functional of (NSE) 1 is and the Nehari manifold is Moreover, we define the least energy of (NSE) and (NSE) 1 by m a := inf M a I a and m ∞ = m 1 := inf M 1 I 1 , respectively. We recall some known results about (NSE) and (NSE) 1 ; see [8].  (iii) u nu -k j=1 w j (·y j n ) → 0 as n → ∞; (iv) I(u n ) = I(u) + k j=1 I ∞ (w j ) + o n (1). Moreover, we agree that in the case k = 0 the above holds without w j .
As in the proof of Lemma 3.1, any (PS) d sequence of I is bounded in H 1 (R 3 ). Then from Lemma 3.5 we easily infer the following two propositions.

Existence of a ground state
In this section, we suppose that (a 2 ) holds. Proof First we assume that (1.5) holds.
Let w a be a ground state of (NSE). Then w a ∈ M a and I a (w a ) = m a . By (1.5) and (2.3), we have that Then w a ∈ Θ, and so there exists t > 0 such that tw a ∈ M. Note that Hence, c ≤ I(tw a ) = 1 4 t 2 w a 2 = t 2 m a < m 1 .
Next we assume (1.6) holds. Let w be the ground state of (NSE) 1 . By (1.6), we get w ∈ Θ. Then there exists t > 0 such that tw ∈ M. First we show that t < 1. Indeed, if t ≥ 1, using w 2 = |w| 4 4 we infer that

Existence of a bound state
Throughout this section we suppose that (a 3 ) holds. First we show that, under (a 3 ), problem (SP) cannot be solved by minimization.

Lemma 5.1 c = m 1 and c is not attained.
Proof First we show that c ≥ m 1 . For all u ∈ U, by the standard argument, there exists t 0 > 0 such that t 0 u ∈ M 1 . By K ≥ 0 and a(x) ≤ 1, we have Using (3.4), we get m 1 ≤ inf u∈U max t>0 I(tu) = c.
Below we shall show that c ≤ m ∞ . Indeed, set w n (x) = w(xz n ), where w the positive solution centered at zero of (NSE) 1 , and |z n | → ∞ as n → ∞.
In fact, since w n 0 in H 1 (R 3 ), by Lemma 2.1(4) we have that so for n large enough, we have and then Considering u n = t n w n ∈ M, we claim that Since t n w n ∈ M, we get a(x)w 4 n dx.
By the previous lemma, we can only hope to find critical points of I at a level higher than m 1 . Similar to [8, Lemma 6.2], we have Lemma 5.2 Functional I satisfies the (PS) d condition for all d ∈ (m 1 , 2m 1 ).